\(R\) ત્રિજ્યા ધરાવતા અલગ અલગ પદાર્થમાથી બનેલા સમતલ અંતર્ગોળ અને સમતલ બહિર્ગોળ લેન્સને આકૃતિમાં દર્શાવ્યા પ્રમાણે મૂકેલા છે.પહેલા લેન્સનો વક્રીભવનાંક \(\mu_1\) અને બીજા લેન્સનો વક્રીભવનાંક \(\mu_2\) હોયતો તંત્રની સંયુક્ત કેન્દ્રલંબાઇ કેટલી થશે?

A \(\frac{R}{{2\left( {{\mu _1} - {\mu _2}} \right)}}\)
B \(\frac{2R}{{\left( {{\mu _1} - {\mu _2}} \right)}}\)
C \(\frac{R}{{\left( {{\mu _1} - {\mu _2}} \right)}}\)
D \(\frac{R}{{2 - \left( {{\mu _1} - {\mu _2}} \right)}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{R}{{\left( {{\mu _1} - {\mu _2}} \right)}}\)

Step-by-step Solution

Detailed explanation

For \(1^{\text {st }}\) lens \(\frac{1}{{{f_1}}} = \left( {\frac{{{\mu _1} - 1}}{1}} \right)\) \(\left( {\frac{1}{\infty } - \frac{1}{{ - R}}} \right) = \frac{{{\mu _1} - 1}}{R}\) For \(2^{n d}\) lens \(\frac{1}{{{f_2}}} = \left( {\frac{{{\mu _2} - 1}}{1}} \right)\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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